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Fixed Point Theory and Applications

, 2010:621469 | Cite as

Some Fixed Point Theorems on Ordered Metric Spaces and Application

  • Ishak Altun
  • Hakan Simsek
Open Access
Research Article

Abstract

We present some fixed point results for nondecreasing and weakly increasing operators in a partially ordered metric space using implicit relations. Also we give an existence theorem for common solution of two integral equations.

Keywords

Continuous Function Integral Equation Point Theorem Existence Theorem Contractive Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Existence of fixed points in partially ordered sets has been considered recently in [1], and some generalizations of the result of [1] are given in [2, 3, 4, 5, 6]. Also, in [1] some applications to matrix equations are presented, in [3, 4] some applications to periodic boundary value problem and to some particular problems are, respectively, given. Later, in [6] O'Regan and Petruşel gave some existence results for Fredholm and Volterra type integral equations. In some of the above works, the fixed point results are given for nondecreasing mappings.

We can order the purposes of the paper as follows.

First, we give a slight generalization of some of the results of the above papers using an implicit relation in the following way.

In [1, 3], the authors used the following contractive condition in their result, there exists Open image in new window such that

Afterwards, in [2], the authors used the nonlinear contractive condition, that is,

where Open image in new window is anondecreasing function with Open image in new window for Open image in new window , instead of (1.1). Also in [2], the authors proved a fixed point theorem using generalized nonlinear contractive condition, that is,

for Open image in new window , where Open image in new window is as above. In the Section 3, we generalized the above contractive conditions using the implicit relation technique in such a way that

for Open image in new window , where Open image in new window is a function as given in Section 2. We can obtain various contractive conditions from (1.4). For example, if we choose

in (1.4), then, we have (1.3). Similarly we can have the contractive conditions in [7, 8, 9] from (1.4).

In some of the above mentioned theorems, the fixed point results are given for nondecreasing mappings. Also in these theorems the following condition is used:

In Section 4, we give some examples such that two weakly increasing mappings need not be nondecreasing. Therefore, we give a common fixed point theorem for two weakly increasing operators in partially ordered metric spaces using implicit relation technique. Also we did not use the condition (1.6) in this theorem. At the end, to see the applicability of our result, we give an existence theorem for common solution of two integral equations using a result of the Section 4.

2. Implicit Relation

Implicit relations on metric spaces have been used in many articles. See for examples, [10, 11, 12, 13, 14, 15].

Let Open image in new window denote the nonnegative real numbers, and let Open image in new window be the set of all continuous functions Open image in new window satisfying the following conditions:

Open image in new window is nonincreasing in variables Open image in new window ;

implies Open image in new window ;

Open image in new window , for all Open image in new window .

Example 2.1.

Open image in new window where Open image in new window , Open image in new window , Open image in new window .

Let Open image in new window and Open image in new window . If Open image in new window then Open image in new window which implies Open image in new window , a contradiction. Thus Open image in new window and Open image in new window . Similarly, let Open image in new window and Open image in new window then Open image in new window If Open image in new window , then Open image in new window Thus Open image in new window is satisfied with Open image in new window . Also Open image in new window , for all Open image in new window . Therefore, Open image in new window .

Example 2.2.

Open image in new window , where Open image in new window .

Let Open image in new window and Open image in new window . If Open image in new window then Open image in new window which is a contradiction. Thus Open image in new window and Open image in new window Similarly, let Open image in new window and Open image in new window then we have Open image in new window If Open image in new window then Open image in new window Thus Open image in new window is satisfied with Open image in new window Also Open image in new window for all Open image in new window Therefore, Open image in new window .

Example 2.3.

Open image in new window where Open image in new window is right continuous and Open image in new window , Open image in new window for Open image in new window

Let Open image in new window and Open image in new window If Open image in new window then Open image in new window which is a contradiction. Thus Open image in new window and Open image in new window Similarly, let Open image in new window and Open image in new window then we have Open image in new window If Open image in new window , then Open image in new window Thus Open image in new window is satisfied with Open image in new window Also Open image in new window , for all Open image in new window Therefore, Open image in new window .

Example 2.4.

Open image in new window , where Open image in new window , Open image in new window , Open image in new window and Open image in new window

Let Open image in new window and Open image in new window Then Open image in new window Similarly, let Open image in new window and Open image in new window then we have Open image in new window If Open image in new window then Open image in new window Thus Open image in new window is satisfied with Open image in new window . Also Open image in new window , for all Open image in new window . Therefore, Open image in new window .

3. Fixed Point Theorem for Nondecreasing Mappings

We need the following lemma for the proof of our theorems.

Lemma 3.1 (See [16]).

Let Open image in new window be a right continuous function such that Open image in new window for every Open image in new window , then Open image in new window , where Open image in new window denotes the Open image in new window -times repeated composition of Open image in new window with itself.

Theorem 3.2.

Let Open image in new window be a partially ordered set and suppose that there is a metric Open image in new window on Open image in new window such that Open image in new window is a complete metric space. Suppose Open image in new window is a nondecreasing mapping such that for all Open image in new window with Open image in new window ,

hold. If there exists an Open image in new window with Open image in new window , then Open image in new window has a fixed point.

Proof.

If Open image in new window , then the proof is finished; so suppose Open image in new window . Now let Open image in new window for Open image in new window . Notice that, since Open image in new window and Open image in new window is nondecreasing, we have
Now since Open image in new window , we can use the inequality (3.1) for these points, then we have
If we continue this procedure, we can have
and so from Lemma 3.1,
Next we show that Open image in new window is a Cauchy sequence. Suppose it is not true. Then we can find a Open image in new window and two sequence of integers Open image in new window with
We may also assume
by choosing Open image in new window to be the smallest number exceeding Open image in new window for which (3.11) holds. Now (3.9), (3.11), and (3.12) imply
Also since
we have from (3.9) that
On the other hand, since Open image in new window , we can use the condition (3.1) for these points. Therefore, we have
Now letting Open image in new window and using (3.14), we have, by continuity of Open image in new window that

From Open image in new window , we have Open image in new window . Therefore, letting Open image in new window in (3.16), we have Open image in new window This is a contradiction since Open image in new window for Open image in new window Thus Open image in new window is a Cauchy sequence in Open image in new window so there exists an Open image in new window with Open image in new window .

If (3.2) holds, then clearly Open image in new window . Now suppose (3.3) holds. Suppose Open image in new window . Now since Open image in new window , then from (3.3), Open image in new window for all Open image in new window . Using the inequality (3.1), we have
so letting Open image in new window from the last inequality, we have

which is a contradiction to Open image in new window . Thus Open image in new window and so Open image in new window .

Remark 3.3.

Note that if we take that

implies Open image in new window ,

Instead of Open image in new window in Theorem 3.2, again we can have the same result.

If we combine Theorem 3.2 with Example 2.1, we obtain the following result.

Corollary 3.4.

Let Open image in new window be a partially ordered set and suppose that there is a metric Open image in new window on Open image in new window such that Open image in new window is a complete metric space. Suppose Open image in new window is a nondecreasing mapping such that for all Open image in new window with Open image in new window ,

hold. If there exists an Open image in new window with Open image in new window , then Open image in new window has a fixed point.

Remark 3.5.

Theorem Open image in new window of [2] follows from Example 2.3, Remark 3.3, and Theorem 3.2.

Remark 3.6.

We can have some new results from other examples and Theorem 3.2.

Remark 3.7.

In Theorem Open image in new window [1], it is proved that if

and hence Open image in new window has a unique fixed point. If condition (3.27) fails, it is possible to find examples of functions Open image in new window with more than one fixed point. There exist some examples to illustrate this fact in [3].

4. Fixed Point Theorem for Weakly Increasing Mappings

Now we give a fixed point theorem for two weakly increasing mappings in ordered metric spaces using an implicit relation. Before this, we will define an implicit relation for the contractive condition of the theorem.

Let Open image in new window be the set of all continuous functions Open image in new window satisfying Open image in new window and the following conditions:

implies Open image in new window ;

Open image in new window and Open image in new window , for all Open image in new window .

We can easily show that, all functions in the Examples in Section 2 are in Open image in new window .

Definition 4.1 (See [17, 18]).

Let Open image in new window be a partially ordered set. Two mappings Open image in new window are said to be weakly increasing if Open image in new window and Open image in new window for all Open image in new window .

Note that, two weakly increasing mappings need not be nondecreasing.

Example 4.2.

Let Open image in new window endowed with usual ordering. Let Open image in new window defined by

then it is obvious that Open image in new window and Open image in new window for all Open image in new window . Thus Open image in new window and Open image in new window are weakly increasing mappings. Note that both Open image in new window and Open image in new window are not nondecreasing.

Example 4.3.

Let Open image in new window be endowed with the coordinate ordering, that is, Open image in new window and Open image in new window . Let Open image in new window be defined by Open image in new window and Open image in new window , then Open image in new window and Open image in new window . Thus Open image in new window and Open image in new window are weakly increasing mappings.

Example 4.4.

Thus Open image in new window and Open image in new window are weakly increasing mappings. Note that Open image in new window but Open image in new window , then Open image in new window is not nondecreasing. Similarly, Open image in new window is not nondecreasing.

Theorem 4.5.

Let Open image in new window be a partially ordered set and suppose that there is a metric Open image in new window on Open image in new window such that Open image in new window is a complete metric space. Suppose Open image in new window are two weakly increasing mappings such that for all comparable Open image in new window ,

hold, then Open image in new window and Open image in new window have a common fixed point.

Remark 4.6.

Note that, in this theorem we remove the condition "there exists an Open image in new window with Open image in new window '' of Theorem 3.2. Again we can consider the result of Remark 3.7 for this theorem.

Proof of Theorem 4.5.

First of all we show that if Open image in new window or Open image in new window has a fixed point, then it is a common fixed point of Open image in new window and Open image in new window . Indeed, let Open image in new window be a fixed point of Open image in new window . Now assume Open image in new window . If we use the inequality (4.7), for Open image in new window , we have
which is a contradiction to Open image in new window . Thus Open image in new window and so Open image in new window is a common fixed point of Open image in new window and Open image in new window . Similarly, if Open image in new window is a fixed point of Open image in new window , then it is also a fixed point of Open image in new window . Now let Open image in new window be an arbitrary point of Open image in new window . If Open image in new window , the proof is finished, so assume Open image in new window . We can define a sequence Open image in new window in Open image in new window as follows:
Without loss of generality, we can suppose that the successive terms of Open image in new window are different. Otherwise, we are again finished. Note that since Open image in new window and Open image in new window are weakly increasing, we have
and continuing this process, we have
Now since Open image in new window and Open image in new window are comparable then, we can use the inequality (4.7) for these points then we have
Similarly, since Open image in new window and Open image in new window are comparable, then we can use the inequality (4.7) for these points then we have
Now again using Open image in new window , we have
Therefore, from (4.18) and (4.22), we can have, for all Open image in new window
Thus from Lemma 3.1, we have, since Open image in new window ,
Next we show that Open image in new window is a Cauchy sequence. For this it is sufficient to show that Open image in new window is a Cauchy sequence. Suppose it is not true. Then we can find an Open image in new window such that for each even integer Open image in new window , there exist even integers Open image in new window such that
We may also assumethat
by choosing Open image in new window to be the smallest number exceeding Open image in new window for which (4.26) holds. Now (4.24), (4.26), and (4.27) imply
Also, by the triangular inequality,
Therefore, we get
Also we have
On the other hand, since Open image in new window and Open image in new window are comparable, we can use the condition (4.7) for these points. Therefore, we have
Now, considering (4.29) and (4.31) and letting Open image in new window in the last inequality, we have, by continuity of Open image in new window , that

From Open image in new window , we have Open image in new window . Therefore, letting Open image in new window in (4.32), we have Open image in new window . This is a contradiction since Open image in new window for Open image in new window . Thus Open image in new window is a Cauchy sequence in Open image in new window , so Open image in new window is a Cauchy sequence. Therefore, there exists an Open image in new window with Open image in new window .

If (4.8) or (4.9) hold then clearly Open image in new window . Now suppose (4.10) holds. Suppose Open image in new window . Now since Open image in new window , then from (4.10), Open image in new window for all Open image in new window . Using the inequality (4.7), we have
So letting Open image in new window from the last inequality, we have

which is a contradiction to Open image in new window . Thus Open image in new window and so Open image in new window .

Remark 4.7.

We can have some new results from Theorem 4.5 with some examples for Open image in new window .

For example, we can have the following corollary.

Corollary 4.8.

Let Open image in new window be a partially ordered set and suppose that there is a metric Open image in new window on Open image in new window such that Open image in new window is a complete metric space. Suppose Open image in new window are two weakly increasing mappings such that for all comparable Open image in new window ,

hold, then Open image in new window and Open image in new window have a common fixed point.

Proof.

Let Open image in new window , then it is obvious that Open image in new window . Therefore, the proof is complete from Theorem 4.5.

5. Application

Consider the integral equations

The purpose of this section is to give an existence theorem for common solution of (5.1) using Corollary 4.8. This section is related to those [19, 20, 21, 22].

Let Open image in new window be a partial order relation on Open image in new window .

Theorem 5.1.

Consider the integral equations (5.1).

(i) Open image in new window and Open image in new window Open image in new window are continuous;

(iii)there exist a continuous function Open image in new window and a right continuous and nondecreasing function Open image in new window such that Open image in new window and Open image in new window for Open image in new window , such that

for each Open image in new window and comparable Open image in new window ;

(iv) Open image in new window .

Then the integral equations (5.1) have a unique common solution Open image in new window in Open image in new window .

Proof.

Let Open image in new window with the usual supremum norm, that is, Open image in new window , for Open image in new window . Consider on Open image in new window the partial order defined by

Then Open image in new window is a partially ordered set. Also Open image in new window is a complete metric space. Moreover, for any increasing sequence Open image in new window in Open image in new window converging to Open image in new window , we have Open image in new window for any Open image in new window . Also for every Open image in new window , there exists Open image in new window which is comparable to Open image in new window and Open image in new window [6].

Now from (ii), we have, for all Open image in new window
Thus, we have Open image in new window and Open image in new window for all Open image in new window . This shows that Open image in new window and Open image in new window are weakly increasing. Also for each comparable Open image in new window , we have

Hence Open image in new window for each comparable Open image in new window . Therefore, all conditions of Corollary 4.8 are satisfied. Thus the conclusion follows.

Notes

Acknowledgment

The authors thank the referees for their appreciation, valuable comments, and suggestions.

References

  1. 1.
    Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proceedings of the American Mathematical Society 2004,132(5):1435–1443. 10.1090/S0002-9939-03-07220-4MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Agarwal RP, El-Gebeily MA, O'Regan D: Generalized contractions in partially ordered metric spaces. Applicable Analysis 2008,87(1):109–116. 10.1080/00036810701556151MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005,22(3):223–239. 10.1007/s11083-005-9018-5MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Nieto JJ, Rodríguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Mathematica Sinica 2007,23(12):2205–2212. 10.1007/s10114-005-0769-0MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Nieto JJ, Pouso RL, Rodríguez-López R: Fixed point theorems in ordered abstract spaces. Proceedings of the American Mathematical Society 2007,135(8):2505–2517. 10.1090/S0002-9939-07-08729-1MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    O'Regan D, Petruşel A: Fixed point theorems for generalized contractions in ordered metric spaces. Journal of Mathematical Analysis and Applications 2008,341(2):1241–1252. 10.1016/j.jmaa.2007.11.026MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ćirić LjB: Generalized contractions and fixed-point theorems. Publications Institut Mathématique 1971,12(26):19–26.MathSciNetMATHGoogle Scholar
  8. 8.
    Ćirić LjB: Fixed points of weakly contraction mappings. Publications Institut Mathématique 1976, 20(34): 79–84.MathSciNetMATHGoogle Scholar
  9. 9.
    Ćirić LjB: Common fixed points of nonlinear contractions. Acta Mathematica Hungarica 1998,80(1–2):31–38.MathSciNetMATHGoogle Scholar
  10. 10.
    Altun I, Turkoglu D: Some fixed point theorems for weakly compatible multivalued mappings satisfying an implicit relation. Filomat 2008,22(1):13–21. 10.2298/FIL0801011AMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Altun I, Turkoglu D: Some fixed point theorems for weakly compatible mappings satisfying an implicit relation. Taiwanese Journal of Mathematics 2009,13(4):1291–1304.MathSciNetMATHGoogle Scholar
  12. 12.
    Imdad M, Kumar S, Khan MS: Remarks on some fixed point theorems satisfying implicit relations. Radovi Matematički 2002,11(1):135–143.MathSciNetMATHGoogle Scholar
  13. 13.
    Popa V: A general coincidence theorem for compatible multivalued mappings satisfying an implicit relation. Demonstratio Mathematica 2000,33(1):159–164.MathSciNetMATHGoogle Scholar
  14. 14.
    Sharma S, Deshpande B: On compatible mappings satisfying an implicit relation in common fixed point consideration. Tamkang Journal of Mathematics 2002,33(3):245–252.MathSciNetMATHGoogle Scholar
  15. 15.
    Turkoglu D, Altun I: A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying an implicit relation. Sociedad Matemática Mexicana. Boletín 2007,13(1):195–205.MathSciNetMATHGoogle Scholar
  16. 16.
    Matkowski J: Fixed point theorems for mappings with a contractive iterate at a point. Proceedings of the American Mathematical Society 1977,62(2):344–348. 10.1090/S0002-9939-1977-0436113-5MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Dhage BC: Condensing mappings and applications to existence theorems for common solution of differential equations. Bulletin of the Korean Mathematical Society 1999,36(3):565–578.MathSciNetMATHGoogle Scholar
  18. 18.
    Dhage BC, O'Regan D, Agarwal RP: Common fixed point theorems for a pair of countably condensing mappings in ordered Banach spaces. Journal of Applied Mathematics and Stochastic Analysis 2003,16(3):243–248. 10.1155/S1048953303000182MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ahmad B, Nieto JJ: The monotone iterative technique for three-point second-order integrodifferential boundary value problems with Open image in new window-Laplacian. Boundary Value Problems 2007, 2007:-9.Google Scholar
  20. 20.
    Cabada A, Nieto JJ: Fixed points and approximate solutions for nonlinear operator equations. Journal of Computational and Applied Mathematics 2000,113(1–2):17–25. 10.1016/S0377-0427(99)00240-XMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ćirić LjB, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory and Applications 2008, 2008:-11.Google Scholar
  22. 22.
    Nieto JJ: An abstract monotone iterative technique. Nonlinear Analysis: Theory, Methods & Applications 1997,28(12):1923–1933. 10.1016/S0362-546X(97)89710-6MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© I. Altun and H. Simsek. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and ArtsKirikkale UniversityYahsihanTurkey

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